Simplify; express your answer in exponential form. Assume $t\neq 0, a\neq 0$. $\dfrac{{(t^{-3}a^{-4})^{-3}}}{{t^{5}a^{-5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{-3}a^{-4})^{-3} = (t^{-3})^{-3}(a^{-4})^{-3}}$ On the left, we have ${t^{-3}}$ to the exponent ${-3}$ . Now ${-3 \times -3 = 9}$ , so ${(t^{-3})^{-3} = t^{9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{-3}a^{-4})^{-3}}}{{t^{5}a^{-5}}} = \dfrac{{t^{9}a^{12}}}{{t^{5}a^{-5}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{9}a^{12}}}{{t^{5}a^{-5}}} = \dfrac{{t^{9}}}{{t^{5}}} \cdot \dfrac{{a^{12}}}{{a^{-5}}} = t^{{9} - {5}} \cdot a^{{12} - {(-5)}} = t^{4}a^{17}$